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# E they are rational 2 beliefs have to be consistent

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Unformatted text preview: about the other players’ actions is not assumed to be correct, but rather, simply constrained by rationality. Deﬁnition A belief of player i about the other players’ actions is a probability measure σ−i ∈ ∏j �=i Σj (recall that Σj denotes the set of probability measures over Sj , the set of actions of player j ). 18 Game Theory: Lecture 3 Mixed Strategy Equilibrium Rationality Rationality imposes two requirements on strategic behavior: (1) Players maximize with respect to some beliefs about opponent’s behavior (i.e., they are rational). (2) Beliefs have to be consistent with other players being rational, and being aware of each other’s rationality, and so on (but they need not be correct). Rational player i plays a best response to some belief σ−i . Since i thinks j is rational, he must be able to rationalize σ−i by thinking every action of j with σ−i (sj ) > 0 must be a best response to some belief j has. . . . Leads to an inﬁnite regress: “I am playing strategy σ1 because I think player 2 is using σ2 , which is a reasonable belief because I would play it if I were � player 2 and I thought player 1 was using σ1 , which is a reasonable thing to � is a b est response to σ � , . . .. expect for player 2 because σ1 2 19 Game Theory: Lecture 3 Mixed Strategy Equilibrium Example Consider the game (from [Bernheim 84]), a1 a2 a3 a4 b1 0, 7 5, 2 7, 0 0, 0 b2 2, 5 3, 3 2, 5 0, −2 b3 7, 0 5, 2 0, 7 0, 0 b4 0, 1 0, 1 0, 1 10, −1 There is a unique Nash equilibrium (a2 , b2 ) in this game, i.e., the strategies a2 and b2 rationalize each other. Moreover, the strategies a1 , a3 , b1 , b3 can also be rationalized: Row will play a1 if Column plays b3 . Column will play b3 if Row plays a3 . Row will play a3 if Column plays b1 . Column will play b1 if Row plays a1 . However b4 cannot be rationalized, and since no rational player will play b4 , a4 can not be rationalized. 20 Game Theory: Lecture 3 Mixed Strategy Equilibrium Never-Best Response Strategies Example Consider the following game: Q X F Q 4, 2 1, 1 3, 0 F 0, 3 1, 0 2, 2 It can be seen that F can be rationalized. If player 1 believes that player 2 will play F, then playing F is rational for player 1, etc. However, playing X is never a best response, regardless of what strategy is chosen by the other player, since playing F always results in better payoﬀs. A strictly dominated strategy will never be a best response, regardless of a player’s beliefs about the other players’ actions. 21 Game Theory: Lecture 3 Mixed Strategy Equilibrium Never-Best Response and S...
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